What is "Word Sense Disambiguation"?
"Word Sense Disambiguation" refers to the idea that words can have different meanings in different contexts. Here are some examples


"I went to the river river bank" vs "I deposited my check at the bank"
"He's mad good at that game" vs "I am so mad at you"


How it effects NLP, comes down to the way we process text. This generally includes the steps of tokenization and embedding them into some form of vector space. These embeddings in many cases are trained either through some self supervised task on some corpora (examples include Word2Vec or Glove) or doing it from scratch under whichever task/data-set is being used.  

Now regarding that paper it does not solve the problem but it does extend a new methodology that assists in helping achieve a better learned generalizable representation for this task. The way I interpreted it is that they don't just use a sense-label (which would be a one-hot encoding / what they call discrete) but instead use a continuous sense representation and do the comparison there. This difference allows for words with similar but different senses to not be equidistant from words with completely different senses.